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Have a proof re. cyclic groups, need a little explaining

  • Thread starter bennyska
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Homework Statement


Let a,b be elements of a group G. show that if ab has finite order, then ba has finite order.


Homework Equations





The Attempt at a Solution


provided proof:
Let n be the order of ab so that (ab)n = e. Multiplying this equation on the left by b and on the right by a, we find that (ba)n+1 = bea = (ba)e. Cancellation of the first factor ba from both sides shows that (ba)n = e, so the order of ba is < n. If the order of ba were less than n, a symmetric argument would show that the order of ab is less than n, contrary to our choice of n. Thus ba has order n also.

okay. so, this is from a chapter on cyclic groups, so i'm assuming that it has to do with, duh, cyclic groups. i know cyclic groups are abelian, and if that were to be assumed from the beginning, i believe this problem would be relatively easy. although, actually, it probably wouldn't even be a problem, since it would be true from the beginning (since we could just say ab=ba, and abn=ban

so:
Let n be the order of ab so that (ab)n = e.
got that, that makes sense, i believe that's the definition of order of an element.
WAIT. here's the crux, something i (maybe) just realized as i typed this sentence. since we define the order of ab such that abn=e, does that imply that G is cyclic, and thus abelian? that would make things easier, if i'm right. so far, a quick internet search has failed me on an answer.
 

Answers and Replies

  • #2
Dick
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If the group is abelian then ab=ba and there is really nothing to prove. The proof you gave holds regardless of whether the group is abelian or not.
 
  • #3
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okay, i was just confused as how b(ab)na could turn into (ba)n+1 without it being abelian. i see how it's from letting (ab)n = e. thanks.
 
  • #4
Dick
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okay, i was just confused as how b(ab)na could turn into (ba)n+1 without it being abelian. i see how it's from letting (ab)n = e. thanks.
It's actually just from regrouping the terms.
 
  • #5
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yeah, actually, i still don't get it. how can you regroup the terms if it's not commutative? or at least it's not assumed to be commutative.
we have (ab)(ab)...(ab) n times. so b(ab)na = b(ab)(ab)...(ab)a...
Oh! got it! thanks!
 

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